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Theorem opth 4075
Description: The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
opth1.1 𝐴 ∈ V
opth1.2 𝐵 ∈ V
Assertion
Ref Expression
opth (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opth
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opth1.1 . . . 4 𝐴 ∈ V
2 opth1.2 . . . 4 𝐵 ∈ V
31, 2opth1 4074 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
41, 2opi1 4070 . . . . . . 7 {𝐴} ∈ ⟨𝐴, 𝐵
5 id 19 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
64, 5syl5eleq 2177 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
7 oprcl 3654 . . . . . 6 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
86, 7syl 14 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
98simprd 113 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 ∈ V)
103opeq1d 3636 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
1110, 5eqtr3d 2123 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
128simpld 111 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13 dfopg 3628 . . . . . . . 8 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1412, 2, 13sylancl 405 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1511, 14eqtr3d 2123 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐵}})
16 dfopg 3628 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
178, 16syl 14 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
1815, 17eqtr3d 2123 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}})
19 prexg 4049 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → {𝐶, 𝐵} ∈ V)
2012, 2, 19sylancl 405 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐵} ∈ V)
21 prexg 4049 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → {𝐶, 𝐷} ∈ V)
228, 21syl 14 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐷} ∈ V)
23 preqr2g 3619 . . . . . 6 (({𝐶, 𝐵} ∈ V ∧ {𝐶, 𝐷} ∈ V) → ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷}))
2420, 22, 23syl2anc 404 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷}))
2518, 24mpd 13 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐵} = {𝐶, 𝐷})
26 preq2 3526 . . . . . . 7 (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷})
2726eqeq2d 2100 . . . . . 6 (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
28 eqeq2 2098 . . . . . 6 (𝑥 = 𝐷 → (𝐵 = 𝑥𝐵 = 𝐷))
2927, 28imbi12d 233 . . . . 5 (𝑥 = 𝐷 → (({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) ↔ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)))
30 vex 2625 . . . . . 6 𝑥 ∈ V
312, 30preqr2 3621 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥)
3229, 31vtoclg 2682 . . . 4 (𝐷 ∈ V → ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
339, 25, 32sylc 62 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐵 = 𝐷)
343, 33jca 301 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐶𝐵 = 𝐷))
35 opeq12 3632 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
3634, 35impbii 125 1 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1290  wcel 1439  Vcvv 2622  {csn 3452  {cpr 3453  cop 3455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461
This theorem is referenced by:  opthg  4076  otth2  4079  copsexg  4082  copsex4g  4085  opcom  4088  moop2  4089  opelopabsbALT  4097  opelopabsb  4098  ralxpf  4597  rexxpf  4598  cnvcnvsn  4922  funopg  5063  funinsn  5078  brabvv  5711  xpdom2  6603  xpf1o  6616  djuf1olem  6801  enq0ref  7055  enq0tr  7056  mulnnnq0  7072  eqresr  7436  cnref1o  9196  fisumcom2  10895  qredeu  11420
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